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Chapter4 6 ½ –

Cube and Cube Root

What is a Cube? So, the cube of a rational number is equal to
the cube of its numerator divided by the cube
A number multiplied by itself two times of its denominator.
results in a cube. æ 2ö 3
Example 1. Find ç ÷
A cube is equal to a number raised to power 3. è 3ø
3
2 ´ 2 ´ 2 = 2 = 8 æ 2ö 3 2 3 2 ´ 2 ´ 2 8
Solution: ç ÷ = = =
8 is a cube equal to 2 raised to power 3. è 3ø 3 3 3 ´ 3 ´ 3 27
or ( )2 3 = , 8 cube of 2 is 8. 3
æ 3
- ö
Example 2. Find ç ÷
Perfect cube è 7 ø
A natural number (n) is said to be a perfect æ 3 3 ( -3) 3
- ö
Solution: ç ÷ =
cube if (n = m3) it is the cube of some natural è 7 ø 7 3
number (m). - ( 3 ) ´ - ) ´ - )
(
3
(
3
=
For example: 7 ´ 7 ´ 7
-27
3
3
3
3
3
,
,
,
1 = 1 2 = 8 3 = 27 4 = 64, 5 = 125, etc. =
343
Thus 1, 8, 27, 64, 125, etc. are per fect cubes.
5. The sum of the cubes of the first n
1. Cubes of all even numbers are even.
natural numbers is equal to the square
3
6 = 6 ´ 6 ´ 6 = 216 of the sum of those numbers.
6 and 216 both are even numbers.
For example: Sum of Cubes of first five
2. Cubes of all odd natural numbers are nat u ral num bers =1 3 + 2 3 + 3 3 + 4 3 + 5 3
odd. 1 + 8 + 27 + 64 + 125 = 225
3
7 = 7 ´ 7 ´ 7 = 343 Square of the sum of the first five natural
7 and 343 both are natural odd numbers. numbers (1 + 2 + 3 + 4 + ) 5 2
2
3. Cube of negative numbers: (15 ) = 225
Cube of negative numbers is negative.
How to know or check a number is a
3
We have, (-1 ) = - ´ - ´ - = -1 perfect cube?
1
1
1
-1 is the cube of itself. The straight method is to check through
3
2
2
Similarly, (-2 ) = - ´ - ´ - = -8 factorization. If the prime factors of a number
2
-8 is the cube of -2. are grouped in triples of same factors, then
that number is called a perfect cube.
4. Cube of a rational number
m In order to check whether a number is a
Let be a rational number(m, n are non zero perfect cube or not, we find its prime factors
n and group together triplets of the prime
integers) then the cube of a is defined as:
3 3 factors. If no factor is left out, then the number
æ mö m m m m is a perfect cube.
ç ÷ = ´ ´ =
è n ø n n n n 3
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Mathematics In Focus - 8